Global Solutions near Homogeneous Steady States in a Multidimensional Population Model with Both Predator- and Prey-Taxis
Mario Fuest
Abstract
We study the system $ (*) \Big\{\!\!\begin{array}{c} u_t = D_1 \Delta u - \chi_1 \nabla \cdot (u \nabla v) + u(\lambda_1 - \mu_1 u + a_1 v) \\ v_t = D_2 \Delta v + \chi_2 \nabla \cdot (v \nabla u) + v(\lambda_2 - \mu_2 v - a_2 u) \end{array} $ (inter alia) for $D_1, D_2, \chi_1, \chi_2, \lambda_1, \lambda_2, \mu_1, \mu_2, a_1, a_2 > 0$ in smooth, bounded domains $\Omega \subset \mathbb R^n$, $n \in \{1, 2, 3\}$. Without any further restrictions on these parameters, we prove that there exists a constant stable steady state $(u_\star, v_\star) \in [0, \infty)^2$, meaning that there is $\varepsilon > 0$ such that if $u_0, v_0 \in W^{2, 2}(\Omega)$ are nonnegative with $\partial_\nu u_0 = \partial_\nu v_0 = 0$ in the sense of traces and $ \|u_0 - u_\star\|_{W^{2,2}(\Omega)} + \|v_0 - v_\star\|_{W^{2,2}(\Omega)} < \varepsilon, $ then there exists a global classical solution $(u, v)$ of $(*)$ with initial data $u_0, v_0$ converging to $(u_\star, v_\star)$ in $W^{2, 2}(\Omega)$. Moreover, the convergence rate is exponential, except for the case $\lambda_2 \mu_1 = \lambda_1 a_2$, where it is is only algebraical. To the best of our knowledge, this constitutes the first global existence result for $(*)$ in the biologically most relevant two- and three-dimensional settings. In the proof, we make use of the special structure in $(*)$ and carefully balance the doubly cross-diffusive interaction therein. Indeed, we introduce certain functionals and combine them in a way allowing for cancellations of the most worrisome terms.